Monotonicity of entropy under bending of Fuchsian representations

Prove that the entropy H(ρ_t) increases monotonically with t for the one-parameter family of representations ρ_t: π₁(S) → PSL₂(C) obtained by bending a Fuchsian representation ρ₀ along the measured geodesic lamination tλ, for 0 ≤ t ≤ t₀, where t₀ is the first parameter value at which ρ_{t₀} ceases to be quasi-Fuchsian.

Background

The paper proves that bending a Fuchsian representation strictly increases entropy for every non-Fuchsian quasi-Fuchsian representation. In the rank-two setting, entropy agrees with both translation-length entropy and the Hausdorff dimension of the limit set.

The authors formulate a stronger parameterized assertion: entropy should vary monotonically along the bending ray obtained by scaling the measured bending lamination. The proposed statement is unresolved up to the first parameter for which the representation leaves the quasi-Fuchsian locus, and is identified as a conjecture to be addressed in future work.